Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1656,2,Mod(1,1656)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1656.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1656, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1656.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-4,0,0,0,0,0,-2,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.2232265747\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 1656.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{5} +3.12311 q^{11} +0.438447 q^{13} -5.12311 q^{17} -3.12311 q^{19} +1.00000 q^{23} -1.00000 q^{25} -3.56155 q^{29} -2.43845 q^{31} +8.24621 q^{37} +9.80776 q^{41} -8.00000 q^{43} +0.684658 q^{47} -7.00000 q^{49} -2.00000 q^{53} -6.24621 q^{55} -10.2462 q^{59} -4.24621 q^{61} -0.876894 q^{65} +3.12311 q^{67} -13.5616 q^{71} -14.6847 q^{73} +3.12311 q^{79} -14.2462 q^{83} +10.2462 q^{85} -11.3693 q^{89} +6.24621 q^{95} +11.3693 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 2 q^{11} + 5 q^{13} - 2 q^{17} + 2 q^{19} + 2 q^{23} - 2 q^{25} - 3 q^{29} - 9 q^{31} - q^{41} - 16 q^{43} - 11 q^{47} - 14 q^{49} - 4 q^{53} + 4 q^{55} - 4 q^{59} + 8 q^{61} - 10 q^{65}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.12311 0.941652 0.470826 − 0.882226i \(-0.343956\pi\)
0.470826 + 0.882226i \(0.343956\pi\)
\(12\) 0 0
\(13\) 0.438447 0.121603 0.0608017 − 0.998150i \(-0.480634\pi\)
0.0608017 + 0.998150i \(0.480634\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.12311 −1.24254 −0.621268 − 0.783598i \(-0.713382\pi\)
−0.621268 + 0.783598i \(0.713382\pi\)
\(18\) 0 0
\(19\) −3.12311 −0.716490 −0.358245 − 0.933628i \(-0.616625\pi\)
−0.358245 + 0.933628i \(0.616625\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.56155 −0.661364 −0.330682 − 0.943742i \(-0.607279\pi\)
−0.330682 + 0.943742i \(0.607279\pi\)
\(30\) 0 0
\(31\) −2.43845 −0.437958 −0.218979 − 0.975730i \(-0.570273\pi\)
−0.218979 + 0.975730i \(0.570273\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.24621 1.35567 0.677834 − 0.735215i \(-0.262919\pi\)
0.677834 + 0.735215i \(0.262919\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.80776 1.53172 0.765858 − 0.643010i \(-0.222315\pi\)
0.765858 + 0.643010i \(0.222315\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.684658 0.0998677 0.0499338 − 0.998753i \(-0.484099\pi\)
0.0499338 + 0.998753i \(0.484099\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −6.24621 −0.842239
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.2462 −1.33394 −0.666972 − 0.745083i \(-0.732410\pi\)
−0.666972 + 0.745083i \(0.732410\pi\)
\(60\) 0 0
\(61\) −4.24621 −0.543672 −0.271836 − 0.962344i \(-0.587631\pi\)
−0.271836 + 0.962344i \(0.587631\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.876894 −0.108765
\(66\) 0 0
\(67\) 3.12311 0.381548 0.190774 − 0.981634i \(-0.438900\pi\)
0.190774 + 0.981634i \(0.438900\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.5616 −1.60946 −0.804730 − 0.593641i \(-0.797690\pi\)
−0.804730 + 0.593641i \(0.797690\pi\)
\(72\) 0 0
\(73\) −14.6847 −1.71871 −0.859355 − 0.511380i \(-0.829134\pi\)
−0.859355 + 0.511380i \(0.829134\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 3.12311 0.351377 0.175688 − 0.984446i \(-0.443785\pi\)
0.175688 + 0.984446i \(0.443785\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −14.2462 −1.56372 −0.781862 − 0.623451i \(-0.785730\pi\)
−0.781862 + 0.623451i \(0.785730\pi\)
\(84\) 0 0
\(85\) 10.2462 1.11136
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.3693 −1.20515 −0.602573 − 0.798064i \(-0.705858\pi\)
−0.602573 + 0.798064i \(0.705858\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.24621 0.640848
\(96\) 0 0
\(97\) 11.3693 1.15438 0.577190 − 0.816610i \(-0.304149\pi\)
0.577190 + 0.816610i \(0.304149\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1656.2.a.j.1.2 2
3.2 odd 2 184.2.a.e.1.2 ✓ 2
4.3 odd 2 3312.2.a.t.1.1 2
12.11 even 2 368.2.a.i.1.1 2
15.2 even 4 4600.2.e.o.4049.2 4
15.8 even 4 4600.2.e.o.4049.3 4
15.14 odd 2 4600.2.a.s.1.1 2
21.20 even 2 9016.2.a.w.1.1 2
24.5 odd 2 1472.2.a.u.1.1 2
24.11 even 2 1472.2.a.p.1.2 2
60.59 even 2 9200.2.a.br.1.2 2
69.68 even 2 4232.2.a.o.1.2 2
276.275 odd 2 8464.2.a.bd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.e.1.2 ✓ 2 3.2 odd 2
368.2.a.i.1.1 2 12.11 even 2
1472.2.a.p.1.2 2 24.11 even 2
1472.2.a.u.1.1 2 24.5 odd 2
1656.2.a.j.1.2 2 1.1 even 1 trivial
3312.2.a.t.1.1 2 4.3 odd 2
4232.2.a.o.1.2 2 69.68 even 2
4600.2.a.s.1.1 2 15.14 odd 2
4600.2.e.o.4049.2 4 15.2 even 4
4600.2.e.o.4049.3 4 15.8 even 4
8464.2.a.bd.1.1 2 276.275 odd 2
9016.2.a.w.1.1 2 21.20 even 2
9200.2.a.br.1.2 2 60.59 even 2